The Logistic Growth SDE
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Transcript of The Logistic Growth SDE
The Logistic Growth SDE
Motivation In population biology the logistic growth model is one of
the simplest models of population dynamics.
To begin studying Stochastic Differential Equations (SDE) we begin by studying the effects of adding a stochastic term to well known deterministic models.
Terminology Stochastic Process: Let I denote an arbitrary nonempty index
set and let Ω,U, P denote a probability space. A family of Rn – valued random variables is a stochastic process.
Markov Property: with probability 1.
; iX t I
( ( ) | ([ , ]) ( ( ) | ( ))oP X t B U t s P X t B X s
Terminology Diffusion Process: A Markov process with continuous sample
paths such that its probability density function satisfies for any
a(x,t) is the infinitesimal mean and is called the drift vector and B(x,t) is the infinitesimal variance and is called the diffusion matrix.
0
0
2
0
1( ) lim ( , ; , ) 0
1( ) lim ( ) ( , ; , ) ( , )
1( ) lim ( ) ( , ; , ) ( , )
y xt
y xt
y xt
i p y t t x t dyt
ii y x p y t t x t dy a x tt
iii y x p y t t x t dy B x tt
0 and ( , )x
Terminology Wiener process: a stochastic process where W(t) depends
continuously on t, and the following hold:( ) ( , )W t
1 2 2 1
2 1
1 2 2 1
1 0
( )For 0 , ( ) ( ) is normally distributed with mean zero and variance
( )For 0 , the increments ( ) ( ) and( ) ( ) are independent
( ) Prob (0) 0 1
i t t W t W tt t
ii t t W t W tW t W t
iii W
Ito’s IntegralStochastic dynamics yields differential equations of the form
where is Gaussian white noise. The goal is to transform (1) into an integral equation and solve
for X(t).
0 0
( ) (0) ( ( ), ) ( , ( )) (2)t t
sX t X f X s s ds g s X s ds
( ) ( ( ), ) ( ( ), ) ( ) (1)X t f X t t dt g X t t t
( )t
Ito’s IntegralThe second integral in (2) is undefined. It can be shown that the
Wiener Process is the derivative of the white noise term.
Using (3) in (2)
0
( ) ( ) or ( ) ( ) (3)t
W t s ds dW t t dt
0 0
(0) ( ( ), ) ( ( ), ) ( ) (4)t t
tX X f X s s ds g X s s dW s
Ito’s Integral The first integral in (4) is the deterministic term and is a
regular integral. The second integral in (4) is the stochastic term and must be
definedTake
We want to define the integral:
( ( ), ) ( ) (5).g X t t W t
0
( ) ( ) ( ) (6).t
t
X t W s dW s
Ito’s IntegralTo examine the behavior of (6) we start by assuming it to be
Riemann-Stieltjes integral and integrating. This yields
The partial sums are defined as
they converge with finer partitions and arbitrary choice of the intermediate points .
0
2 20( ) ( )( ) (7)
2
t
t
W t W tW s ds
11
( )( ( ) ( )) (8)n
n i i ii
S W W t W t
i
Ito’s IntegralThe approximation sums converge in mean square. They can be
written as
Convergence depends on the choice of the intermediate point.
Choose .
1
21
1 1
( ) ( ).i i
n n
t t i ii i
E W W t
1i it
Ito’s IntegralEquation (6) then becomes
By convention an Ito SDE is written as
and satisfies the integral equation
0
2 20 0( ) ( )( ) ( ) .
2 2
t
t
W t W t t tW s dW s
( ) ( ( ), ) ( ( ), ) ( )dX t f X t t dt g X t t dW t
0 0
( ) (0) ( ( ), ) ( ( ), ) ( ).t t
X t X f X s s ds g X s s dW s
Ito’s FormulaSuppose ( ) is a solution to the following Ito SDE:
( ) ( ( ), ) ( ( ), ) ( ).
If ( , ) is a real-valued function defined for
and t [a,b], with continuous partial derivatives, ,
, and
X tdX t f X t t dt g X t t dW t
F x t xFt
Fx
R
2
2
2 2
2
, then
( ( ), ) ( ( ), ) ( ( ), ) ( ) where
( , ) ( , ) ( , ) ( , )( , ) ( , )2
( , )( , ) ( , )
Fx
dF X t t f X t t dt g X t t dW t
F x t F x t g x t F x tf x t f x tt x x
F x tg x t g x tx
Example: Exponential Growth Consider the SDE ,exponential growth with environmental variation, where c and r
are positive constants.
Let Applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:
( ) ( ) ( ) ( )dX t rX t dt cX t dW t
( , ) ln( ).F x t x
2
( ) (0)exp([ ]) ( )).2cX t X r t cW t
Example: Logistic Growth Consider the SDE
logistic growth with environmental variation, where c, K and r are positive constants.
Let Applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:
( )( ) ( ) 1 ( ) ( ),X tdX t rX t dt cX t dW tK
1( , ) .xF x t
2
2
12
0
exp([ ]) ( ))2( ) .
(0) exp([ ]) ( ))t
crK
cr t cW tX t
X r s cW s ds
Example: Bimodal Equations Consider the SDE
logistic growth with environmental variation, where c, K and r are positive constants.
Let . Then applying Ito’s formula and integrating from 0 to t, and solving for X(t) yields:
2 ( )( ) ( ) 1 ( ) ( )X tdX t rX t dt cX t dW tK
21( , )x
F x t
2
2
2
12
0
exp([ ]) ( ))2( ) .
(0) exp([ ]) ( ))t
crK
cr t cW tX t
X r s cW s ds
Graphs: The Basic Equations
Graphs: ( ) ( ) ( , ) ( )dX t X t dt x t dW t
Graphs: ( )( ) ( )(1 ) ( , ) ( )X tKdX t rX t dt x t dW t
Graphs: 2 ( )( ) ( )(1 ) ( , ) ( )X tKdX t rX t dt x t dW t
Considerations Effects of the coefficient of the stochastic
term. How to determine the correct coefficients for a specific problem
Expected Gaussian versus graphed Levy distribution
References An Introduction to Stochastic Process with Applications to
Biology Linda J.S. Allen
Stochastic Differential Equations Ludwig Arnold
Introduction to Stochastic Differential Equations Thomas Gard